Maximizing the Geometric Mean of User-Rates to Improve Rate-Fairness: Proper vs. Improper Gaussian Signaling

This paper considers a reconfigurable intelligent surface (RIS)-aided network, which relies on a multiple antenna array aided base station (BS) and an RIS for serving multiple single antenna downlink users. To provide reliable links to all users over the same bandwidth and same time-slot, the paper proposes the joint design of linear transmit beamformers and the programmable reflecting coefficients of an RIS to maximize the geometric mean (GM) of the users' rates. A new computationally efficient alternating descent algorithm is developed, which is based on closed-forms only for generating improved feasible points of this nonconvex problem. We also consider the joint design of widely linear transmit beamformers and the programmable reflecting coefficients to further improve the GM of the users' rates. Hence another alternating descent algorithm is developed for its solution, which is also based on closed forms only for generating improved feasible points. Numerical examples are provided to demonstrate the efficiency of the proposed approach.